Ricerc@Sapienza

This article defines the class of -valued autoregressive (AR) processes with a unit root of finite type, where is a possibly infinite-dimensional separable Hilbert space, and derives a generalization of the Granger–Johansen Representation Theorem valid for any integration order . An existence theorem shows that the solution of an AR process with a unit root of finite type is necessarily integrated of some finite integer order d, displays a common trends representation with a finite number of common stochastic trends, and it possesses an infinite-dimensional cointegrating space when .

In kernel methods, temporal information on the data is commonly included by using time-delayed embeddings as inputs. Recently, an alternative formulation was proposed by defining a gamma-filter explicitly in a reproducing kernel Hilbert space, giving rise to a complex model where multiple kernels operate on different temporal combinations of the input signal.